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A new set of $ h(1) \oplus su(2)$ vector algebra eigenstates on the matrixdomain is obtained by defining them as eigenstates of a generalizedannihilation operator formed from a linear combination of the generators ofthis algebra which eigenvalues are distributed as the elements of a squarecomplex normal matrix. A combined method is used to compute these eigenstates,namely, the method of exponential operators and that of a system of first-orderlinear differential equations. We compute these states for all possiblecombination of generators and classify them in different categories accordingto a generalized commutation relation as well as according to the value of acharacteristic parameter related to the $su(2)$ algebra eigenvalues. Proceedingin this way, we found a subset of generalized vector coherent states in thematrix domain which can be easily separated from the general set ofSchr\"odinger-Robertson minimum uncertainty intelligent states. In particular,for a special choice of the matrix eigenvalue parameters we found the so-calledvector coherent states with matrices associated to the Heisenberg-Weyl group aswell as a generalized version of them, and also a direct connection with thecoherent state quantization of quaternions.

$h(1) \oplus su(2)$ vector algebra eigenstates with eigenvalues in the matrix domain